NAG Library Routine Document
F08WPF (ZGGEVX)
1 Purpose
F08WPF (ZGGEVX) computes for a pair of n by n complex nonsymmetric matrices A,B the generalized eigenvalues and, optionally, the left and/or right generalized eigenvectors using the QZ algorithm.
Optionally it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.
2 Specification
SUBROUTINE F08WPF ( |
BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO) |
INTEGER |
N, LDA, LDB, LDVL, LDVR, ILO, IHI, LWORK, IWORK(*), INFO |
REAL (KIND=nag_wp) |
LSCALE(N), RSCALE(N), ABNRM, BBNRM, RCONDE(*), RCONDV(*), RWORK(6*N) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), ALPHA(N), BETA(N), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK)) |
LOGICAL |
BWORK(*) |
CHARACTER(1) |
BALANC, JOBVL, JOBVR, SENSE |
|
The routine may be called by its
LAPACK
name zggevx.
3 Description
A generalized eigenvalue for a pair of matrices A,B is a scalar λ or a ratio α/β=λ, such that A-λB is singular. It is usually represented as the pair α,β, as there is a reasonable interpretation for β=0, and even for both being zero.
The right generalized eigenvector
vj corresponding to the generalized eigenvalue
λj of
A,B satisfies
The left generalized eigenvector
uj corresponding to the generalized eigenvalue
λj of
A,B satisfies
where
ujH is the conjugate-transpose of
uj.
All the eigenvalues and, if required, all the eigenvectors of the complex generalized eigenproblem
Ax=λBx, where
A and
B are complex, square matrices, are determined using the
QZ algorithm. The complex
QZ algorithm consists of three stages:
- A is reduced to upper Hessenberg form (with real, non-negative subdiagonal elements) and at the same time B is reduced to upper triangular form.
- A is further reduced to triangular form while the triangular form of B is maintained and the diagonal elements of B are made real and non-negative. This is the generalized Schur form of the pair
A,B
.
This routine does not actually produce the eigenvalues
λj, but instead returns
αj and
βj such that
The division by
βj becomes your responsibility, since
βj may be zero, indicating an infinite eigenvalue.
- If the eigenvectors are required they are obtained from the triangular matrices and then transformed back into the original coordinate system.
For details of the balancing option, see
Section 3 in F08WVF (ZGGBAL).
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Wilkinson J H (1979) Kronecker's canonical form and the
QZ algorithm
Linear Algebra Appl. 28 285–303
5 Parameters
- 1: BALANC – CHARACTER(1)Input
On entry: specifies the balance option to be performed.
- BALANC='N'
- Do not diagonally scale or permute.
- BALANC='P'
- Permute only.
- BALANC='S'
- Scale only.
- BALANC='B'
- Both permute and scale.
Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does. In the absence of other information, BALANC='B' is recommended.
Constraint:
BALANC='N', 'P', 'S' or 'B'.
- 2: JOBVL – CHARACTER(1)Input
On entry: if
JOBVL='N', do not compute the left generalized eigenvectors.
If JOBVL='V', compute the left generalized eigenvectors.
Constraint:
JOBVL='N' or 'V'.
- 3: JOBVR – CHARACTER(1)Input
On entry: if
JOBVR='N', do not compute the right generalized eigenvectors.
If JOBVR='V', compute the right generalized eigenvectors.
Constraint:
JOBVR='N' or 'V'.
- 4: SENSE – CHARACTER(1)Input
On entry: determines which reciprocal condition numbers are computed.
- SENSE='N'
- None are computed.
- SENSE='E'
- Computed for eigenvalues only.
- SENSE='V'
- Computed for eigenvectors only.
- SENSE='B'
- Computed for eigenvalues and eigenvectors.
Constraint:
SENSE='N', 'E', 'V' or 'B'.
- 5: N – INTEGERInput
On entry: n, the order of the matrices A and B.
Constraint:
N≥0.
- 6: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the matrix A in the pair A,B.
On exit:
A has been overwritten. If
JOBVL='V' or
JOBVR='V' or both, then
A contains the first part of the Schur form of the ‘balanced’ versions of the input
A and
B.
- 7: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08WPF (ZGGEVX) is called.
Constraint:
LDA≥max1,N.
- 8: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the matrix B in the pair A,B.
On exit:
B has been overwritten.
- 9: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F08WPF (ZGGEVX) is called.
Constraint:
LDB≥max1,N.
- 10: ALPHA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 11: BETA(N) – COMPLEX (KIND=nag_wp) arrayOutput
On exit:
ALPHAj/BETAj, for
j=1,2,…,N, will be the generalized eigenvalues.
Note: the quotients ALPHAj/BETAj may easily overflow or underflow, and BETAj may even be zero. Thus, you should avoid naively computing the ratio αj/βj. However, maxαj will always be less than and usually comparable with A2 in magnitude, and maxβj will always be less than and usually comparable with B2.
- 12: VL(LDVL,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VL
must be at least
max1,N if
JOBVL='V', and at least
1 otherwise.
On exit: if
JOBVL='V', the left generalized eigenvectors
uj are stored one after another in the columns of
VL, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have
real part+imag. part=1.
If
JOBVL='N',
VL is not referenced.
- 13: LDVL – INTEGERInput
On entry: the first dimension of the array
VL as declared in the (sub)program from which F08WPF (ZGGEVX) is called.
Constraints:
- if JOBVL='V', LDVL≥ max1,N ;
- otherwise LDVL≥1.
- 14: VR(LDVR,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VR
must be at least
max1,N if
JOBVR='V', and at least
1 otherwise.
On exit: if
JOBVR='V', the right generalized eigenvectors
vj are stored one after another in the columns of
VR, in the same order as the corresponding eigenvalues. Each eigenvector will be scaled so the largest component will have
real part+imag. part=1.
If
JOBVR='N',
VR is not referenced.
- 15: LDVR – INTEGERInput
On entry: the first dimension of the array
VR as declared in the (sub)program from which F08WPF (ZGGEVX) is called.
Constraints:
- if JOBVR='V', LDVR≥ max1,N ;
- otherwise LDVR≥1.
- 16: ILO – INTEGEROutput
- 17: IHI – INTEGEROutput
On exit:
ILO and
IHI are integer values such that
Aij=0 and
Bij=0 if
i>j and
j=1,2,…,ILO-1 or
i=IHI+1,…,N.
If BALANC='N' or 'S', ILO=1 and IHI=N.
- 18: LSCALE(N) – REAL (KIND=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the left side of
A and
B.
If
plj is the index of the row interchanged with row
j, and
dlj is the scaling factor applied to row
j, then:
- LSCALEj = plj , for j=1,2,…,ILO-1;
- LSCALE = dlj , for j=ILO,…,IHI;
- LSCALE = plj , for j=IHI+1,…,N.
The order in which the interchanges are made is
N to
IHI+1, then
1 to
ILO-1.
- 19: RSCALE(N) – REAL (KIND=nag_wp) arrayOutput
On exit: details of the permutations and scaling factors applied to the right side of
A and
B.
If
prj is the index of the column interchanged with column
j, and
drj is the scaling factor applied to column
j, then:
- RSCALEj=prj, for j=1,2,…,ILO-1;
- if
RSCALE=drj, for j=ILO,…,IHI;
- if
RSCALE=prj, for j=IHI+1,…,N.
The order in which the interchanges are made is
N to
IHI+1, then
1 to
ILO-1.
- 20: ABNRM – REAL (KIND=nag_wp)Output
On exit: the 1-norm of the balanced matrix A.
- 21: BBNRM – REAL (KIND=nag_wp)Output
On exit: the 1-norm of the balanced matrix B.
- 22: RCONDE(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDE
must be at least
max1,N.
On exit: if
SENSE='E' or
'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array.
If
SENSE='N' or
'V',
RCONDE is not referenced.
- 23: RCONDV(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
RCONDV
must be at least
max1,N.
On exit: if
SENSE='V' or
'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array.
If
SENSE='N' or
'E',
RCONDV is not referenced.
- 24: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 25: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08WPF (ZGGEVX) is called.
If
LWORK=-1, a workspace query is assumed; the routine only calculates the optimal size of the
WORK array, returns this value as the first entry of the
WORK array, and no error message related to
LWORK is issued.
Suggested value:
for optimal performance,
LWORK must generally be larger than the minimum; increase workspace by, say,
nb×N, where
nb is the optimal
block size.
Constraints:
- if SENSE='N', LWORK≥max1,2×N;
- if SENSE='E', LWORK≥max1,4×N;
- if SENSE='B' or 'V', LWORK≥max1,2×N×N+2×N.
- 26: RWORK(6×N) – REAL (KIND=nag_wp) arrayWorkspace
Real workspace.
- 27: IWORK(*) – INTEGER arrayWorkspace
-
Note: the dimension of the array
IWORK
must be at least
max1,N+2.
If
SENSE='E',
IWORK is not referenced.
- 28: BWORK(*) – LOGICAL arrayWorkspace
-
Note: the dimension of the array
BWORK
must be at least
max1,N.
If
SENSE='N',
BWORK is not referenced.
- 29: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO=1 to N
The QZ iteration failed. No eigenvectors have been calculated, but ALPHAj and BETAj should be correct for j=INFO+1,…,N.
- INFO=N+1
Unexpected error returned from
F08XSF (ZHGEQZ).
- INFO=N+2
Error returned from
F08YXF (ZTGEVC).
7 Accuracy
The computed eigenvalues and eigenvectors are exact for a nearby matrices
A+E and
B+F, where
and
ε is the
machine precision.
An approximate error bound on the chordal distance between the
ith computed generalized eigenvalue
w and the corresponding exact eigenvalue
λ
is
An approximate error bound for the angle between the
ith computed eigenvector
VLi
or
VRi
is given by
For further explanation of the reciprocal condition numbers
RCONDE and
RCONDV, see Section 4.11 of
Anderson et al. (1999).
Note: interpretation of results obtained with the
QZ algorithm often requires a clear understanding of the effects of small changes in the original data. These effects are reviewed in
Wilkinson (1979), in relation to the significance of small values of
αj and
βj. It should be noted that if
αj and
βj are
both small for any
j, it may be that no reliance can be placed on
any of the computed eigenvalues
λi=αi/βi. You are recommended to study
Wilkinson (1979) and, if in difficulty, to seek expert advice on determining the sensitivity of the eigenvalues to perturbations in the data.
8 Further Comments
The total number of floating point operations is proportional to n3.
The real analogue of this routine is
F08WBF (DGGEVX).
9 Example
This example finds all the eigenvalues and right eigenvectors of the matrix pair
A,B,
where
and
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix pair is used.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.
9.1 Program Text
Program Text (f08wpfe.f90)
9.2 Program Data
Program Data (f08wpfe.d)
9.3 Program Results
Program Results (f08wpfe.r)